Cells respond to their environment by processes known as signal transduction. A signal transduction process generally begins with a hormone, pheromone or neurotransmitter binding to its receptor, which in turn activates a cascade of biochemical events culminating in a cellular response. In a biological system (e.g., a cell), there can be many different types of cascades and each one of these cascades can be considered to be a signaling module which, when activated, leads to a cellular response. Cells have many different kinds of receptors and are faced with many different kinds of signals, and multiple modules can be activated and/or active at the same time. Often, these modules interact with each other such that the net cellular response is an integrative function of all the inputs (e.g., cascades) interacting with the cell's unique machinery, which can lead to a range of cell-specific responses. The complexity of signal transduction networks poses a challenge to traditional experimental analysis. Mathematical models that can help to explore and critically evaluate data are needed to deal with this biological complexity (Weng et al., (1999) Science 284: 92–96 and Bhalla et al., (1999) Science 283: 381–387).
The quantitative analysis of the behavior of enzymes using Michaelis-Menten kinetics assumes that the enzymes are in an aqueous three-dimensional space. However cells are not well-stirred sacs, and to quantitatively analyze cellular regulatory pathways it can be important to take into consideration the local concentration and time-dependent diffusion of second messengers and protein cascades (e.g., spatio-temporal effects).
To develop a more refined quantitative approach to this problem, partial differential equations are more adequate than ordinary differential equations. Biochemical components can be described in a point-to-point way, with a time-dependent evolution for description of signaling processes, metabolic processes, transport processes, and other biological processes.
Pointwise descriptions of the various components of signaling present a higher order of complexity with respect to global, average, and bulk descriptions in several regards. One aspect of the complexity is the difficulty in solving partial differential equations, either theoretically or numerically. Another aspect of the complexity is that pointwise descriptions naturally involve the geometry of the space in which these processes are analyzed.
In implementing partial differential equations that provide spatial and temporal resolution of the pathway, the highly complex geometry of a cell can be addressed mathematically. Some of the features that are implemented in the present invention include a novel mathematical treatment of the complex geometry of the cells via homogenization and concentrated capacity, as well as the coupled diffusion of second messengers with explicit spatial and temporal control using partial differential equations.